 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2008/b/5/a/solution

lo g S

S_{t} = s exp {(μ - \frac{1}{2} σ^{2}) t + σ W_{t}} .

E e^{a W_{t}} = e^{a^{2} t /2}

then yields

E S_{t}^{2} = s^{2} e^{(2 μ - σ^{2}) t} E e^{2 σ W_{t}} = s^{2} e^{(2 μ + σ^{2}) t} .

This expectation is under the physical measure and therefore uses drift

μ

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